Online TCP Acknowledgment Under General Delays

Abstract

In a seminal work, Dooly, Goldman, and Scott (STOC 1998; JACM 2001) introduced the classic Online TCP Acknowledgment problem. In this problem, a sequence of n packets arrives over time, and the objective is to minimize both the number of acknowledgments sent and the total delay experienced by the packets. They showed that a natural greedy algorithm, which acknowledges when the delay of pending packets equals the acknowledgment cost, is 2-competitive.
Online TCP Acknowledgment is the canonical online problem with delay, capturing the fundamental tradeoff between reducing service cost through batching and the delay incurred by pending requests. Prior work has largely focused on richer service-cost models, e.g., Joint Replenishment and Multi-Level Aggregation. However, other than the work of Albers and Bals (SODA 2003), which studies maximum delay and closely related objectives, not much is known about general delay costs beyond the sum of delay costs of requests.
In this work, we study Online TCP Acknowledgment under two generalized delay-cost models that we call batch-aware and batch-oblivious. In the batch-aware model, each batch incurs a delay cost that depends on the packet delays within that batch. For the max-over-batches objective, which generalizes Albers and Bals, we show that greedy remains 2-competitive for every monotone batch-delay function. For the sum-over-batches objective, the picture changes sharply: greedy is Ω(n)-competitive, and the optimal deterministic competitive ratio is Θ(log n). Our matching upper bound requires only the minimal assumption that the batch delay function is monotone.
In the batch-oblivious model, the delay cost is a function of the global packet-delay vector. We show that greedy is 2-competitive for continuous submodular delay costs, and more generally under a weaker zero-coordinate diminishing-marginals condition. This yields 2-competitive algorithms for 𝓁_p norms, Top-k norms, and ordered norms. Using the submodular-norm approximation of Patton, Russo, and Singla, we also obtain an O(log n)-competitive algorithm for arbitrary symmetric norms.